There is loop hole if you are taking one of the disciplines that is only offered once a year...you get four years instead of two (but still only four attempts).
So one approval equals to 4 attempts.
a question, what if fail one discipline after 4 attempts, then apply for another discipline immediately? for example, electrical, there are 3 options, and whichever you pass, the roster will show "ELE", and there won't be much difference between electronics and power, especially people from power most likely took a lot of electronics courses in college.
That does not appear to be an open loophole. The rules say nothing about which exam you take, only that you have 4 "consecutive" attempts. See page 48 of the link:
"© An applicant who has been denied a license for failure to pass the examination on the principles and practice of engineering with
prescribed time may not re-apply for a license until one (1) year has passed from the date on the notification of failure to pass the
examination or until the applicant has completed at least six (6) additional semester hours of formal college level classroom courses
relevant to the applicant’s dominant branch or discipline of experience. "
http://engineers.texas.gov/downloads/law_rules_122512.pdf